Average Error: 29.4 → 0.1
Time: 4.8s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12879.754469549705 \lor \neg \left(x \le 12427.129501950301\right):\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{\frac{1}{x}}{x}, \frac{-3}{x}\right) - 3 \cdot \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -12879.754469549705 \lor \neg \left(x \le 12427.129501950301\right):\\
\;\;\;\;\mathsf{fma}\left(-1, \frac{\frac{1}{x}}{x}, \frac{-3}{x}\right) - 3 \cdot \frac{1}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}\\

\end{array}
double f(double x) {
        double r132609 = x;
        double r132610 = 1.0;
        double r132611 = r132609 + r132610;
        double r132612 = r132609 / r132611;
        double r132613 = r132609 - r132610;
        double r132614 = r132611 / r132613;
        double r132615 = r132612 - r132614;
        return r132615;
}
↓
double f(double x) {
        double r132616 = x;
        double r132617 = -12879.754469549705;
        bool r132618 = r132616 <= r132617;
        double r132619 = 12427.129501950301;
        bool r132620 = r132616 <= r132619;
        double r132621 = !r132620;
        bool r132622 = r132618 || r132621;
        double r132623 = -1.0;
        double r132624 = 1.0;
        double r132625 = r132624 / r132616;
        double r132626 = r132625 / r132616;
        double r132627 = 3.0;
        double r132628 = -r132627;
        double r132629 = r132628 / r132616;
        double r132630 = fma(r132623, r132626, r132629);
        double r132631 = 1.0;
        double r132632 = 3.0;
        double r132633 = pow(r132616, r132632);
        double r132634 = r132631 / r132633;
        double r132635 = r132627 * r132634;
        double r132636 = r132630 - r132635;
        double r132637 = r132616 - r132624;
        double r132638 = r132616 * r132637;
        double r132639 = r132616 + r132624;
        double r132640 = r132639 * r132639;
        double r132641 = r132638 - r132640;
        double r132642 = r132616 * r132616;
        double r132643 = r132624 * r132624;
        double r132644 = r132642 - r132643;
        double r132645 = r132641 / r132644;
        double r132646 = r132622 ? r132636 : r132645;
        return r132646;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -12879.754469549705 or 12427.129501950301 < x

    1. Initial program 59.2

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]

    2. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{-\left(1 \cdot \frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)}\]

    3. Simplified0.3

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)}\]
    4. Using strategy rm

    5. Applied fma-udef0.3

      \[\leadsto \frac{-1}{{x}^{2}} - \color{blue}{\left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)}\]

    6. Applied associate--r+0.3

      \[\leadsto \color{blue}{\left(\frac{-1}{{x}^{2}} - 3 \cdot \frac{1}{x}\right) - 3 \cdot \frac{1}{{x}^{3}}}\]

    7. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{\frac{1}{x}}{x}, \frac{-3}{x}\right)} - 3 \cdot \frac{1}{{x}^{3}}\]

    if -12879.754469549705 < x < 12427.129501950301

    1. Initial program 0.1

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
    2. Using strategy rm

    3. Applied frac-sub0.1

      \[\leadsto \color{blue}{\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{\left(x + 1\right) \cdot \left(x - 1\right)}}\]

    4. Simplified0.1

      \[\leadsto \frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{\color{blue}{x \cdot x - 1 \cdot 1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12879.754469549705 \lor \neg \left(x \le 12427.129501950301\right):\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{\frac{1}{x}}{x}, \frac{-3}{x}\right) - 3 \cdot \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))